---
tag: 6BB0KD6752
kind: concept
name: Birkhoff's axioms
msc: [51M05, 51F05]
sources:
  - url: https://en.wikipedia.org/wiki/Birkhoff%27s_axioms
same_as:
  wikidata: Q3625630
  wikipedia: Birkhoff's axioms
---

## Summary

Birkhoff's axioms are four postulates for plane [[Euclidean geometry|KRZD96GSVC]] that George David Birkhoff gave in 1932.
They take the [[real numbers|08J2M1JYAG]] as known and measure distances with a ruler and angles with a protractor.
Because the real numbers do much of the work, the list is very short, and it became the basis of many school geometry courses.

## Explanations

### L1 human

Hilbert's axioms build geometry from scratch, which makes the list long.
Birkhoff instead allows a ruler and a protractor: every line is a copy of the number line, and every angle has a size in degrees.
With these tools and one axiom about similar triangles, all of plane geometry follows.
For example, the angle sum of a triangle is a short consequence of the similarity axiom.